Zhikov's problem deals with distributional solutions to the stationary drift diffusion problem. For $L^p$ regular divergence-free drifts $b$, it is known in 3D that $H^1$ zero mean solutions $u$ are unique provided $b$ is in at least $L^2$. For $b\in L^p, p<3/2$ at least two solutions are known to exist. I will show that: 1) if divergence-free drift is slightly more regular, namely $b\in W^{1,1}$, then $H^1$ solutions with zero mean are unique, 2) in dimensions $n\geq 4$, if divergence-free $b\in L^p, 1\leq p<\frac{2(n-1)}{n+1}$, then there exist at least two $H^1$ zero mean solutions, 3) for $p<n-1, n\geq 3$ there exist at least two zero mean solutions $u\in L^{p'}\cap W^{1,r},r<\frac{p'}{1+\frac{n-1}{p'}}$, where $p'$ is a Hölder conjugate of $p$.

A talk is based on a common paper with Wojtek Ożański. Part 1) is based on a duality method and the DiPerna-Lions commmutator estimate, parts 2) and 3) are obtained via convex integration construction using the Szekelyhidi-Modena Mikado flows.